On the homotopy fixed points of Maurer-Cartan spaces with finite group actions

Abstract

We develop the basic theory of Maurer-Cartan simplicial sets associated to (shifted complete) L ∞ algebras equipped with the action of a finite group. Our main result asserts that the inclusion of the fixed points of this equivariant simplicial set into the homotopy fixed points is a homotopy equivalence of Kan complexes, provided the L ∞ algebra is concentrated in non-negative degrees. As an application, and under certain connectivity assumptions, we provide rational algebraic models of the fixed and homotopy fixed points of mapping spaces equipped with the action of a finite group.There are other alternative constructions: Getzler's γ • (L), Robert-Nicoud-Vallette's R(L), and Buijs-Félix-Murillo-Tanré's 〈L〉 (see [16,26,9], respectively). We explain in Section 5 that our main results also hold for these smaller alternative simplicial sets. Conventions:In this paper, all the algebraic structures are considered over the field of the rational numbers, but our results hold over any field of characteristic zero. We use a homological grading for L ∞ -algebras. All topological spaces are assumed to be CW-complexes. If V is a graded vector space, and G is a group that acts linearly on V , then the graded vector space of fixed points isand the graded vector space of coinvariants isThe linear dual of a graded vector space V is denoted V ∨ . Acknowledgments:The authors are very grateful to Marc Stephan for teaching them about equivariant homotopy theory, and to the Max Planck Institute for Mathematics in Bonn for its hospitality and financial support.